In 2011, he was found not guilty in the NSW Supreme Court of their murders by reason of mental illness.

An inquest at Sydney's Glebe Coroner's Court heard on Monday that health professionals never detained Anthony under the Mental Health Act (MHA) despite years of "bizarre and often threatening" behaviour.

"At no time ... was he judged to come within the test requiring him to be scheduled under the MHA," counsel assisting the Coroner, Peggy Dwyer, said in her opening address.

She said a major issue at the inquest would be whether Anthony should have been sectioned under the act.

The inquest heard that Anthony, who sometimes spells his name Antony, suffered from paranoid schizophrenia and believed his family was part of a plan targeting him.

It was also told that Anthony had for years suffered delusions and thought neighbours were "tormenting him".

At one time in 2004, the inquest heard Anthony had claimed to hear neighbours talking about him and blowing smoke in his room and had thought he was being filmed and recorded.

On another occasion, he was observed making a stabbing motion towards a neighbour's property.

Ms Dwyer said the deaths of Nick and Chloe had "left a terrible hole" in the Waterlow family.

"They remain devastated and grief stricken by their deaths," she said.

"They are also confounded as to how this would happen to a loving, gifted family, who had tried so hard to get Anthony help for may years."